If one considers Pythagoras' Theorem,
X2 + Y2 = Z2
then we might imagine if this holds true
for powers of 3, or 4, and so on. In other
words, if this is true-
Xn + Yn = Zn, for n > 2
This is what Pierre de Fermat, an amateur
French mathematician said was false. So
he did not believe that such an equation
would be possible with whole numbers,
which are greater than 2.
He started by proving that his theorem was
true, for n = 3. Later on, the mathematician
Leonhard Euler proved this for n = 4.
Then the next big breakthrough came when
Sophie Germain created a set of primes, called
Sophie-Germain primes, which have the
property that when they are doubled, and increased
by one, i.e. 2n + 1, the result is another prime.
She showed how Fermat's Theorem could be
proved for such numbers.
Even though there were so many proofs of
individual cases of Fermat's Last Theorem,
It wouldn't be sufficient to say that the Theorem
is absolutely, undoubtedly, irrefutably true.
Mathematicians struggled with the complete proof
of Fermat's Last Theorem, but although the mystery
remained unsolved, the effort to solve it created
some amazing works of mathematics.
Meanwhile, in Japan, two students, namely Yutaka
Taniyama and Goro Shimura, thought up the
Taniyama-Shimura conjecture, which stated that
every elliptic equation is modular, neither of which I
understand, so I can't explain it here.
The relation between the two came when Gerhard Frey
theorised that if the Taniyama-Shimura conjecture was
true, it would consecutively prove that Fermat's Last
Theorem was true, as well. This claim was substantiated
by Ken Ribet, who made the connection absolute.
Then after 8 years of effort, English mathematician, Andrew
Wiles, solved Fermat's Last Theorem, finally showing that
the equation
Xn + Yn = Zn, for n > 2,
has no solutions.
The interesting thing is that Fermat, who knew not more maths
than the average school student, claimed he had found a solution
to the problem that was eventually solved after 350 years, with
maths that Fermat could not have known. So, there just might
exist a scribbling on a piece of paper with elementary maths, that
could substitute Wiles' 8 years of hard effort, or it might be up to
us to rediscover that proof, if in fact it exists?
X2 + Y2 = Z2
then we might imagine if this holds true
for powers of 3, or 4, and so on. In other
words, if this is true-
Xn + Yn = Zn, for n > 2
This is what Pierre de Fermat, an amateur
French mathematician said was false. So
he did not believe that such an equation
would be possible with whole numbers,
which are greater than 2.
He started by proving that his theorem was
true, for n = 3. Later on, the mathematician
Leonhard Euler proved this for n = 4.
Then the next big breakthrough came when
Sophie Germain created a set of primes, called
Sophie-Germain primes, which have the
property that when they are doubled, and increased
by one, i.e. 2n + 1, the result is another prime.
She showed how Fermat's Theorem could be
proved for such numbers.
Even though there were so many proofs of
individual cases of Fermat's Last Theorem,
It wouldn't be sufficient to say that the Theorem
is absolutely, undoubtedly, irrefutably true.
Mathematicians struggled with the complete proof
of Fermat's Last Theorem, but although the mystery
remained unsolved, the effort to solve it created
some amazing works of mathematics.
Meanwhile, in Japan, two students, namely Yutaka
Taniyama and Goro Shimura, thought up the
Taniyama-Shimura conjecture, which stated that
every elliptic equation is modular, neither of which I
understand, so I can't explain it here.
The relation between the two came when Gerhard Frey
theorised that if the Taniyama-Shimura conjecture was
true, it would consecutively prove that Fermat's Last
Theorem was true, as well. This claim was substantiated
by Ken Ribet, who made the connection absolute.
Then after 8 years of effort, English mathematician, Andrew
Wiles, solved Fermat's Last Theorem, finally showing that
the equation
Xn + Yn = Zn, for n > 2,
has no solutions.
The interesting thing is that Fermat, who knew not more maths
than the average school student, claimed he had found a solution
to the problem that was eventually solved after 350 years, with
maths that Fermat could not have known. So, there just might
exist a scribbling on a piece of paper with elementary maths, that
could substitute Wiles' 8 years of hard effort, or it might be up to
us to rediscover that proof, if in fact it exists?
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